Self Drawing Equation

1. Have fun
2. Create an 2D equation that draws itself on a cartesian plane

Motivation

What if someone discovers the real equation of the universe comprising of initial conditions? He could write it on a piece of paper and that piece of paper would contain an equation whose solution would contain the piece of paper with the equation itself!

This is a very intriguing concept, and I wanted to see if I could create a 2D equation that would draw itself on a cartesian plane.

Drawing the alphabet

First of all, I needed to concentrate on drawing some letters and numbers i needed for later usage. I used the following alphabet:

0123456789+*-/^=()xyteansigfwhrdlo.b,

I then started a project with Desmos defining an equation for each letter.

Compressing data

Before composing the letter, I needed a target equation to represent. I choose the following model:

"some very long equation" where a = 123... some very long number

For simplicity, I choose to place the right side of the equation starting exactly at x = 0.

Decoding the parameter

I needed a way to encode some information into a numeric variable and then decode it inside the equation that was represented itself.

To do this, I decided to associate a number to each letter and then encode the equation in a number by concatenating the numbers associated to each letter and using it as a base 28 number.

To decode the number, I used the following equation: \left\lfloor \frac{a}{b^{\left\lfloor x \right\rfloor} } \right\rfloor - \left\lfloor \frac{a}{b^{\left\lfloor x - 1 \right\rfloor} } b \right\rfloor Where a is the encoded number, b is the base and x is the position of the digit to decode.

You can see the encoding and decoding in action in the following example:

Composing the equation

TL/DW: just compose the equations in a way that they are "enabled" when the digit is the correct one.

I used Matlab to programmatically compose the equation. The final equation was:

f(1,x,0)+f(atan(((x-floor(x))-0.5)^2*2.75+(y-1.05)^2-0.25),x,1)+f((atan((x-floor(x))-0.5)+4*(sign((y-1.05)^2-0.24)+1))*(atan(y-(x-floor(x))*0.4-1.34)+4*(sign(((x-floor(x))-0.35)^2-0.025)+1))*(atan(y-0.57)+4*(sign(((x-floor(x))-0.5)^2-0.1)+1)),x,2)+f((atan((y-0.57-e^(((x-floor(x))-0.8)*100))*(y-(x-floor(x))-0.43))+4*((sign(((x-floor(x))-0.48)^2-0.087))+1))*(atan(((x-floor(x))-0.5)^2+(y-1.3)^2-0.08)+4*(sign(y+((x-floor(x)))/(5)-1.35)-1)),x,3)+f(atan((((x-floor(x))-0.5)^2*0.6+(y-0.82)^2-0.07)*(((x-floor(x))-0.5)^2*0.6+(y-1.3)^2-0.05))+4*(sign(((x-floor(x))+(y*1.3-1.4)^2-0.45))-1),x,4)+f((atan(y-1.8*(x-floor(x))-0.4)+4*(sign((y-1.2)^2-0.115)+1))*(atan(y-0.85)+4*(sign(((x-floor(x))-0.55)^2-0.08)+1))*(atan((x-floor(x))-0.67)+4*(sign((y-1.05)^2-0.23)+1))*(atan(y-0.56)+4*(sign(((x-floor(x))-0.65)^2-0.03)+1)),x,5)+f((atan(y-1.55)+4*(sign(((x-floor(x))-0.5)^2-0.055)+1))*(atan((x-floor(x))-0.30)+4*(sign((y-1.32)^2-0.055)+1))*(atan(((x-floor(x))-0.5)^2+(y-0.85)^2-0.1)+4*(sign(5*(x-floor(x))-y-0.4)-1)),x,6)+f((atan(((x-floor(x))-0.57)^2+(y-0.84)^2-0.08))*(atan(((x-floor(x))-0.6)^2+((y-1.05)^2)/(2)-0.12)+4*(sign(1.15*(x-floor(x))-y+0.5)+1)),x,7)+f((atan((y+e^((0.2-(x-floor(x)))*100)-1.55)*(y-3*(x-floor(x))+0.9))+4*(sign(((x-floor(x))-0.52)^2-0.09)+sign(0.5-y)+2)),x,8)+f(atan((((x-floor(x))-0.5)^2+(y-0.81)^2-0.075)*(((x-floor(x))-0.5)^2+(y-1.32)^2-0.06)),x,9)+f(atan(((x-floor(x))-0.53)^2+(y-1.26)^2-0.08)*(atan(((x-floor(x))-0.5)^2+((y-0.85-((x-floor(x)))/(3))^2)/(2)-0.1)+4*(sign(y-1.1*(x-floor(x))-0.3)+1)),x,10)+f(atan(((x-floor(x))-0.5)*(y-1.02))+4*(sign(((x-floor(x))-0.5)^2+(y-1.02)^2-0.15)+1),x,11)+f(atan((((abs((x-floor(x))-0.5)))/((y-1.23))-0.01)*(((abs((x-floor(x))-0.5)))/((y-1.23))-4)*(((abs((x-floor(x))-0.5)))/((y-1.23))+0.7))+4*(sign(((x-floor(x))-0.5)^2+(y-1.23)^2-0.1)+1),x,12)+f(atan(y-1)+4*(sign(((x-floor(x))-0.5)^2-0.1)+1),x,13)+f(atan(y-2.1*(x-floor(x)))+4*(sign(((x-floor(x))-0.5)^2-0.1)+1),x,14)+f(atan(y+1.3*abs(((x-floor(x))-0.5))-1.5)+4*(sign(((x-floor(x))-0.5)^2-0.1)+1),x,15)+f(atan((y-1)^2-0.015)+4*(sign(((x-floor(x))-0.5)^2-0.16)+1),x,16)+f(atan(((x-floor(x))-1.5)^2+(y-0.95)^2-0.9)+4*(sign((x-floor(x))-0.8)+1),x,17)+f(atan(((x-floor(x))+0.5)^2+(y-0.95)^2-0.9)+4*(sign((x-floor(x))-0.2)-1),x,18)+f((atan(abs((x-floor(x))-0.5)-abs(y-0.9))+4*(sign(((x-floor(x))-0.5)^2+(y-0.9)^2-0.22)+1))*(atan((y-0.9)^2-0.105)+4*(sign((abs((x-floor(x))-0.5)-0.25)^2-0.01)+1)),x,19)+f((atan(y-1.23)+4*(sign((abs((x-floor(x))-0.52)-0.29)^2-0.01)+1))*(atan(y-2*(x-floor(x))+0.5)+4*(sign(((x-floor(x))-0.62)^2-0.06)+1))*(atan(y+2*(x-floor(x))-1.6)+4*(sign(((x-floor(x))-0.36)^2-0.03)+1))*(atan(y-0.25)+4*(sign(((x-floor(x))-0.35)^2-0.03)+1)),x,20)+f((atan((x-floor(x))-0.34)+4*(sign((y-1.05)^2-0.18)+1))*(atan(y-1.23)+4*(sign(((x-floor(x))-0.45)^2-0.1)+1))*(atan((((x-floor(x))-0.6)^2)/(2)+(y-0.75)^2-0.05)+4*(sign(y-0.63)+1)),x,21)+f((atan(((x-floor(x))-0.5)^2+(y-0.88)^2-0.13)+4*(sign(y-3*(x-floor(x))+1.65)-1))*(atan(y-0.92)+4*(sign(((x-floor(x))-0.5)^2-0.125)+1)),x,22)+f(((((x-floor(x))-0.4)^2)/(e^(x-floor(x)))+(y-0.1*(x-floor(x))-0.69)^2-0.05)*(atan((x-floor(x))-0.73)+4*(sign((y-0.83)^2-0.08)+1))*(atan((((x-floor(x))-0.5)^2)/(1.5)+(y-1.05)^2-0.04)+4*(sign(y-1.11)-1))*(atan(y-0.55)+4*(sign(((x-floor(x))-0.8)^2-0.008)+1)),x,23)+f((atan(((x-floor(x))-0.51)^2-0.07)+4*(sign((y-0.56)*(y+0.3*(x-floor(x))-1.3))+1))*(atan(y-0.57)+4*(sign(((abs((x-floor(x))-0.5)-0.27)^2-0.01))+1))*(atan(y-1.22)+4*(sign(((x-floor(x))-0.2)^2-0.005)+1))*(atan((((x-floor(x))-0.5)^2)/(1.5)+(y-0.95-0.1*(x-floor(x)))^2-0.05)+4*(sign(y-1.05)-1)),x,24)+f((atan(((((x-floor(x))-0.5)^2)/(2)+(y-0.72)^2-0.04)*((((x-floor(x))-0.5)^2)/(2)+(y-1.08)^2-0.03))+4*(sign((y-(x-floor(x))-0.4)*(y-0.9))-1))*(atan(abs((x-floor(x))-0.5)-0.27)+4*(sign((y-0.9*(x-floor(x))-0.45)^2-0.008)+1)),x,25)+f((atan((x-floor(x))-0.5)+4*(sign(((y-1.07)^2-0.27)*((y-1.35)^2-0.01))+1))*(atan(y-0.56)+4*(sign(((x-floor(x))-0.5)^2-0.1)+1))*(atan(y-1.23)+4*(sign(((x-floor(x))-0.4)^2-0.012)+1)),x,26)+f(atan(((x-floor(x))-0.45)^2+((y-0.9)^2)/(1.2)-0.09)*(atan((x-floor(x))+e^((-20*y+4))-0.76)+4*(sign((y-0.75)^2-0.26)+1))*(atan(y-1.23)+4*(sign(((x-floor(x))-0.83)^2-0.01)+1)),x,27)+f((atan((x-floor(x))-0.43)+4*(sign((y-1)^2-0.18)+1))*(atan((y-0.9)^2-0.11)+4*(sign(((x-floor(x))-0.5)^2-0.08)+1))*(atan((((x-floor(x))-0.67)^2)/(2)+(y-1.4)^2-0.03)+4*(sign(y-0.2*(x-floor(x))-1.3)-1)),x,28)+f((atan(y-abs(3*abs((x-floor(x))-0.52)-0.5)-0.55)+4*(sign(((x-floor(x))-0.52)^2-0.155)+1))*(atan(y-1.22)+4*(sign(((abs((x-floor(x))-0.52)-0.35)^2-0.01))+1)),x,29)+f((atan(((x-floor(x))-0.51)^2-0.07)+4*(sign((y-0.56)*(y+0.9*(x-floor(x))-1.76))+1))*(atan(y-0.57)+4*(sign(((abs((x-floor(x))-0.5)-0.27)^2-0.01))+1))*(atan(y-1.55)+4*(sign(((x-floor(x))-0.2)^2-0.005)+1))*(atan((((x-floor(x))-0.5)^2)/(1.5)+(y-0.95-0.1*(x-floor(x)))^2-0.05)+4*(sign(y-1.05)-1)),x,30)+f((atan((x-floor(x))-0.4)+4*(sign(((y-0.9)^2-0.12))+1))*(atan(y-0.56)+4*(sign(((x-floor(x))-0.45)^2-0.1)+1))*(atan(y-1.23)+4*(sign(((x-floor(x))-0.3)^2-0.012)+1))*(atan(5*y-sin(5*(x-floor(x))-2)-5)+4*(sign(((x-floor(x))-0.65)^2-0.07)+1)),x,31)+f((atan(((x-floor(x))-0.47)^2+(y-0.88)^2-0.11))*(atan((x-floor(x))-0.8)+4*(sign((y-1.05)^2-0.26)+1))*(atan(y-0.55)+4*(sign(((x-floor(x))-0.88)^2-0.01)+1))*(atan(y-1.55)+4*(sign(((x-floor(x))-0.72)^2-0.01)+1)),x,32)+f((atan((x-floor(x))-0.5)+4*(sign(((y-1.07)^2-0.27))+1))*(atan(y-0.56)+4*(sign(((x-floor(x))-0.5)^2-0.1)+1))*(atan(y-1.55)+4*(sign(((x-floor(x))-0.4)^2-0.012)+1)),x,33)+f(atan(((x-floor(x))-0.5)^2+(y-0.87)^2-0.13),x,34)+f(atan(sign(((x-floor(x))-0.5)^2+(y-0.6)^2-0.01)+1),x,35)+f((atan((0.53-(x-floor(x)))^2+(y-0.88)^2-0.11))*(atan(0.2-(x-floor(x)))+4*(sign((y-1.05)^2-0.26)+1))*(atan((y-0.55)*(y-1.55))+4*(sign((0.12-(x-floor(x)))^2-0.01)+1)),x,36)+f(atan(sign(((x-floor(x))-0.4)^2+(y-2*(x-floor(x))+0.3)^2-0.01)+1),x,37) = 0 where f(t,x,n)=(sign((g(x)-n)^2-0.1)-1)*t and g(x)=((floor(a/(38^floor(-x)))-floor(a/(38^floor(1-x)))*38)*(-sign(x)+1)/2 + (floor(a/(10^floor(1-x+floor(log(a)/log(10)))))-floor(a/(10^floor(2-x+floor(log(a)/log(10)))))*10+1)*(sign(x*(1+floor(log(a)/log(10))-x))+1)/2)*((x-floor(x))/(x-floor(x))) and a = 40993464515018873593298995509687481446009302941035508945052888583657290600898634951083785548622620422458533085369800081498776522733888145701346924817667976026685586222226184553322757730169533306516580264511220490733783801235435828739899680243456829005077505339856774529519607014858532918031987758197481865082743531605632632115859266439341769566645868000385827388876402588582262838672044594877254504522687751610286479208237591783958880031074246099579305212202201830952549120568367224629206801981657834196181226526042712922462647500606139566860370750939429412416387887919088195349611613282127461212510866473614129025736993709774518993087086595956118482116700119997512410896312493272238393408690557718777176750395034915373207873347199246203861242776914924796269704402186533599775050715166927491541465811690669481750928587907251810314666017366197293228720863315753993613186381714216388994716446899422242938594137883362166442910489955510032716500815209387130986720048392109543588052264406525041895740592424637557394330668445662791161469442114673271413960648420578335810361574467098303235040576110087917846332539518982454846987298777088611580959312534593939265686109746728462449561106084410232251718048290683619474848871195621433460618335676655000694671288030760091570101417232420655468873610804084459137823693992007213799511810651238500108614919005121148310800144401966735338705050394142992330801361467282560366718157977037950753259235996840760906315069483690808666451253197627849643145501775435105401606819261336712445603971958390383488675784064109976260513119034225463601036889942296815522296445635729701481613161896135088542948490265326250625640010485411318045711818823533210163326333610065859285728727767361029393129496571614321871013504680058290411734473298491367630147338414847131581301555596959026366552015479189816585490615851650508199967755445181948581383306234997447700362565300020426874057571739867028876088676554100057348766921113543779254539013054924791093180469947142601553115577879357653005919420422776188724975598275890052475267552578768610880801575511850592270559420842301423529379604160678042902714824353521910976487938848632484501456832215891291208898360133615355199543575216451826522764421037240622348703121643490609666398327633228126939998374018112141653082122705781275942454678102072607312745637414121993284690947367878686241940208881485315976161533423085775459034049298364610080815865743385339228799146357587606643748600525207821576666152140692278616608249275675646543700475193928181517057459927485589609233262331323542909438231944369245952890144484679349121921350734555695251794947053446727206680203400438879324236397341535524263419450538292679062540945121753665513619361031655868429759926731024494633652882774040326430511636632803916720995350026320676719509579071844710309770253474519605765300751133252120334942949820315039579386435949726408589598954550017151435615555889842862351533375611609097186433418535853134562675445076380626168541964700738415348123765202538150582618670764838259644637345605909429257539809358986913439222127077457506923009908022062910167960322366791138154501553548122924732329564130366503988782978941126363670833559032620353220450299805818804541331033946675453878703729826173754736819277238553151562731195937743817977039317212183668070100304262890448697845125972253541783393811133364023517349842345824060208894511990197890938016961761046478401595669246918876045580845829735665075378279017466054221240348928524667759027200599908648951702818947506586752788764035672015593348553475109787510927514900012362614625554121659555149079009778011528065573400405318194668883718900807074757411648264515047493088265848383120424114379422163176551312508900945616825395925084808922317310464200375609508736131448156588265721083411243027399498196973275450658392143173570457866584872393433616249653031219978295235338642094334104360134994580001729857982174069629619354265327437046126801410260269781448950086243447067305361524016105260791897182592743571100647338824436352737946899881801994787314467173361264475868515307236945428771327174536045609284045414983561776048957790992808454453523392233794221042874275432624693251757195599519173226371302874262091246647526128822346136875532700254188317272491010833613844417001679503032377814368641140426973823402557478620349834228340485569999670736911436345075262798951635603357687906492770102863471319470080439698459239355785249273112616648582472735149115601758551792030992668933645708604245121989159705573949083109747131643284990443155219551684614948394398700207180174006355919663950165370576639305110299424723707846952892142131385407933028584762363899475234138390245825655614883368287964434161201985640174927097811716175945381112443273746974712062642411061053346236378312929165177200682790392728147001048318408141400129166353081331282926409557901245134837170089781691386367464807824841458465992076813658101388912723267346033042328337725792543645818865701803828739834660885696182004071921714969045054846184604232283629130640511865944600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Plotting the equation

It is obviously impossible to use a stardard plotter. I made a plotter in matlab but it was extremely stupid and slow, but the result was nice. I never finihed plotting the equation because it was too slow. This is a partial plot I made:

Tupper's self-referential formula

Source code

Having fun with other equations

Unfortunately, I lost nearly all the funny equations I produced in the time.

This is an equation that produces a fractal-like structure (the plot is a little buggy):

(sin(log(((4*x+sqrt(16*x^2-12*(x^2+y^2)))/4)pi)/log(1.25)))(sin(log(((4*x-sqrt(16*x^2-12(x^2+y^2)))/4)*pi)/log(1.25)))=0

This is a time-dependent equation that produces an orbiting planet around a star:

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Un post condiviso da Luca Ciucci (@luca.ciucci99)

This is an equation I made for the PI day:

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Un post condiviso da Luca Ciucci (@luca.ciucci99)

This is an equation I made for my girlfriend's birthday before we started dating:

arctan((x-tan(y*1.2)/10)^6*10^7+y^6-1)*(1/(((x-2)^2+(y-0.4)^2-0.15)*((x-2)^2+(y-0.4)^2-0.2))+1/((x-1.6)^6*10^5+(y+0.4)^6/100)+1/((x-1.75+0.7y)^6*10^5+(y+0.4)^6/20)-1000)*(1/((x-3.6)^6*10^5+(y)^6/1000)+1/((x-3.9+0.3y)^6*10^5+(y)^6/1000)+1/((x-4.2)^6*10^5+(y)^6/1000)-1000)*(1/((x+0.35-0.3y-5)^6*10^5+(y)^6/1000)+1/((-x+0.35-0.3y+5)^6*10^5+(y)^6/1000)+1/((y)^6*10^5+(x-5)^6/5)-1000)*arctan(1/((x-0.7)^6*10^7+y^6*10)+1/10^7/((x-0.95)^6/10+y^6)+1/10^7/((x-0.95)^6/10+(y-0.9)^6)+1/10^7/((x-0.95)^6/10+(y+0.9)^6)-0.1)*arctan(1/((x-2.9)^6*10^7+y^6*10)+1/10^7/((x-3.15)^6/10+y^6)+1/10^7/((x-3.15)^6/10+(y-0.9)^6)+1/10^7/((x-3.15)^6/10+(y+0.9)^6)-0.1)=0

It might not be very beautiful, but I think it was an original present nevertheless ❤️